Mathematical Sciences: Topology

Ruth Charney教授和Michael Davis教授的项目涉及拓扑、几何和几何群论方面的问题。该项目的主要焦点是研究非正曲率的分段欧几里得多面体（在Gromov的CAT(0)不等式的意义上）。理解这些空间的局部几何等价于理解具有分段球面度量的多面体何时是“大”的（即，满足CAT(1)不等式）。这个项目试图回答一些关于“大”分段球形多面体的问题，以及负弯曲黎曼流形何时支持分段双曲CAT(0)度规（与B. Okun）的相关问题。查尼的研究还包括通过几何技术研究Artin群体；Davis的研究包括某些奇异纤维束在流形上光滑环面作用的应用（与R. Scott合作）。群是抽象对称概念的代数结构，在数学和数学物理中普遍存在。从历史上看，几何群论领域涉及了几个数学领域的思想——拓扑学、组合学和群论。最近引入的以微分几何为模式的技术已经导致了该领域令人兴奋的新发展，以及拓扑问题的新应用。其中一种技术，拓扑度量空间的“非正曲率”，在这些发展中发挥了重要作用。查尼和戴维斯的项目旨在更好地理解曲率的概念，以及如何将其应用于拓扑学和群论中的问题。* * *

9505003 Davis The project of Professors Ruth Charney and Michael Davis involves problems in topology, geometry, and geometric group theory. The main focus of the project is the study of piecewise Euclidean polyhedra of nonpositive curvature (in the sense of Gromov's CAT(0) inequality). Understanding the local geometry of such spaces is equivalent to understanding when polyhedra with piecewise spherical metrics are "large" (i.e., satisfy the CAT(1) inequality). This project attempts to answer a number of questions concerning "large" piecewise spherical polyhedra, as well as the related question of when a negatively curved Riemannian manifold supports a piecewise hyperbolic CAT(0) metric (with B. Okun). Charney's research also includes the study of Artin groups via geometric techniques; Davis's includes the study of certain singular fiber bundles with applications to smooth torus actions on manifolds (with R. Scott). Groups are algebraic structures abstracting the concept of symmetry, which is pervasive in mathematics and mathematical physics. The field of geometric group theory has, historically, involved ideas from several areas of mathematics -- topology, combinatorics, and group theory. The recent introduction of techniques patterned on differential geometry has led to exciting new developments in the field as well as new applications to problems in topology. One of these techniques, a version of "nonpositive curvature" for topological metric spaces, has played a major role in these developments. The project of Charney and Davis aims at a better understanding of this notion of curvature and how it may be applied to problems in topology and group theory. ***